Many professional mathematicians wish to contribute to making mathematics more attractive to gifted young people. One way to pursue this goal is to create of problems that require a great deal of common sense, imagination, and, very often, a specific problem solving strategy. This article introduces such a strategy, the principle of the extreme element. Although the name is not widely used, the principle may help you to solve some non-standard mathematical problems. The material is based on my personal experience gained in extracurricular work with mathematically gifted students, and in my participation in mathematics competitions at different levels as a contestant and organiser.